Exact soliton solutions and the significance of time-dependent coefficients in the Boussinesq equation: theory and application in mathematical physics

This article effectively establishes the exact soliton solutions for the Boussinesq model, characterized by time-dependent coefficients, employing the advanced modified simple equation, generalized Kudryashov and modified sine–Gordon expansion methods. The adaptive applicability of the Boussinesq system to coastal dynamics, fluid behavior, and wave propagation enriches interdisciplinary research across hydrodynamics and oceanography. The solutions of the system obtained through these significant techniques make a path to understanding nonlinear phenomena in various fields, surpassing traditional barriers and further motivating research and application. Significant impacts of the coefficients of the equation, wave velocity, and related parameters are evident in the profiles of soliton-shaped waves in both 3D and 2D configurations when all these factors are treated as variables, which are not seen in the case for constant coefficients. This study enhances the understanding of the significant role played by nonlinear evolution equations with time-dependent coefficients through careful dynamic explanations and detailed analyses. This revelation opens up an interesting and challenging field of study, with promising insights that resonate across diverse scientific disciplines.


Analysis of the methods
This section provides a broad overview of the employed methods to tackle the problem of variable coefficient Boussinesq system.In the investigation of wave solutions for the system, we used the power of three distinct yet complementary methods.Generally these techniques provide closed-form wave solutions with further free parameters than the other methods.Therefore, the obtained solutions can accurately explain the phenomenon of the system.Moreover, the chosen methods are able to provide all types of soliton solutions such as kink, periodic, bell-shape, spike type, and other solitons; which are not feasible in many techniques.So each of these approaches has been strategically chosen and meticulously executed to offer unique insights and solutions.Elaborations on the distinctive approaches of the three techniques in finding exact solutions are presented below.

Common starting for the first two methods
Let us consider the evolution equation in the following nonlinear form: wherein F is a polynomial function of u(x, t) and its partial derivatives, incorporating both the highest order derivatives and nonlinear terms.
Consider the form below for the traveling wave transformation.
where both p(t) and q(t) are functions that possess differentiability with respect to t.
Through the application of this wave transformation, Eq. (1) can be transformed into the following nonlinear ordinary differential equation: In here, the prime and dot marks signify the derivatives with respect to ξ and t , respectively, i.e.U ′ = dU dξ , ṗ = dp dt andq = dq dt .
Step 2: By utilizing the balance homogeneity principle between the leading nonlinear terms and the maximum order derivatives present in Eq. (3), we determine the value of the positive integer n appearing in Eq. (4).
Step 3: Inserting the sequential derivatives of U(ξ ) into Eq. ( 3 leads to the creation of a polynomial equation representing ϕ −1 .In the ensuing stages, a system of equations arises through the process of setting the coeffi- cients of x k ϕ −i to zero for k = 0, 1, 2, . . .and i = 0, 1, 2, . . ., n in the polynomial.This system of equations yields a collection of algebraic equations for the parameters a i (t), p(t), q(t) and differential equations involving the derivatives of p(t), q(t) with regard to t.
Step 4: Subsequently, the solution of Eq. ( 3) can be found by first solving the system derived in step 3 to determine ϕ(ξ ), a i (t), p(t) and q(t) , and then substituting these values into Eq. ( 4) ultimately leading to the solutions of Eq. (1).

The Generalized Kudryashov method
Following are the key steps that describe the methodology of the generalized Kudryashov 51-53 : Step 1: The approach proposes that the precise solution for Eq.(1) be represented in the subsequent rational structure: where a i (t)(i = 0, 1, 2, . . ., n) and b j (t)(j = 0, 1, 2, . . ., m) are functions dependent on t, to be solved at a later juncture, under the conditions a n (t) = 0 and b m (t) = 0 .Furthermore, the following ODE is satisfied by Q(ξ ): It is clear that Eq. ( 6) possesses a solution in the following form: Here, the symbol A stands as an integrating constant.
Step 2: The principle of homogeneous balance will guide us to identify the choices of positive integer for n and m within Eq. (5).To illustrate, we achieve balance by equating the highest order derivative with the corresponding highest order nonlinear term presented in Eq. (3).
Step 3: Substituting Eq. ( 5) into Eq.( 3) along with Eq. ( 6), we acquire a polynomial of Q(ξ ) .By setting all of the coefficients of the like powers of Q(ξ ) , which are multiplied by x k for k = 0, 1, 2, . . . to zero, we attain both algebraic and differential systems of equations of parameters a i , b j , p , q and p , q respectively.Through the utili- zation of the Mathematica or Maple software package program, these systems of equations can be solved.This process enables us to compute the values of the unknown parameters a i , b j , p and q .Subsequently, we find the exact solutions for the reduced Eq.(3).

The modified sine-Gordon expansion method
The sine-Gordon equation is given by Herein, u denotes a function of both x and t .By introducing the wave variable ξ = px + qt , we can rewrite the preceding equation as the subsequent nonlinear expression.
Here, the variable U(ξ ) = u(x, t) , where ξ and q/p stand for the amplitude and velocity of the traveling wave, respectively.Now Eq. ( 9) can be expressed as a first order differential equation as follows: The K appears in the above equation as an integrating constant.Substituting U(ξ )/2 = w(ξ ) , a 2 = 1/(p 2 + q 2 ) and K = 0 into Eq.( 10) yields: taking w = w(ξ ) and a = 1.
By solving Eq. ( 11) through separation of variables and simplification, we can confirm the validity of the following two interesting relations: The value of the integrating constant, r = 0.
In light of the assumption, the solution for the polynomial nonlinear wave equation having variable coefficients of the form F(u, u t , u x , u tt , u xt , u xx , . . . ) = 0 can be regarded as: The transformation U(ξ ) = u(x, t) , ξ = p(t)x + q(t) is utilized to reformulate the nonlinear wave equa- tion with time-dependent coefficients, leading to an equivalent nonlinear equation, with p, q, a 0 , a i and b i (i = 1, 2, 3, . . ., n) as functions of time.
Through the application of the results indicated in Eqs. ( 12) and ( 13), we can represent Eq. ( 14) in an alternative form as follows: which represents a polynomial of degree n.
The parameter n can be determined using the balance procedure, which is the comparison of the highest order derivative with the highest nonlinear term.Subsequent to finding the parameter n , simplifying by insert- ing solution (15) into the transformed nonlinear equation yields a trigonometric series in the form of x k sin(jw) and x k cos(jw) , where j = 0, 1, 2, . . ., n and k = 0, 1, 2, . . . 47.By equating to zero all the relevant harmonic terms sin(jw) and cos(jw) in multiplicative form with x k , we establish a set of equations that relate to the unknowns p, q, a 0 , a i and b i .As a result, the problem converts into algebraic equations for p, q, a 0 , a i , b i and differential equa- tions of p and q in relation to time.Computerized calculations are used to ascertain the values of p, q, a 0 , a i , b i and subsequently, the solutions proposed in Eqs. ( 12), ( 13) and ( 15) are applied.

Extraction of solutions
Consider the Boussinesq model subject to the variable coefficients of the following form: with α(t) and β(t) being differentiable functions throughout all t.
The system of equations stated above is utilized to model the bidirectional movement of particular water waves in a smooth, horizontal channel saturated with a liquid that is both inviscid and irrotational 66 .The timedependent coefficients in the model indicate that the properties of the medium or fluid are changing with time.These coefficients can significantly influence the transmission of waves through the medium, leading to multifaceted wave interactions and phenomena.
By implementing the transformations U(ξ ) = u(x, t) and V (ξ ) = v(x, t) , with ξ representing the wave variable defined as ξ = p(t)x + q(t) , and substituting them into Eqs. ( 16 and ( 17), the equations transform as follows: Integrating once Eqs.(18) and (19) with regard to ξ , we obtain By replacing V from Eq. (20) into Eq.( 21) and simplifying thereafter, we derive The utilization of the homogeneous balance principle for the highest order linear term U ′′ and the nonlinear term U 2 in (22) results in (12)  www.nature.com/scientificreports/

Solutions through the modified simple equation method
Since the balance number is n = 2 , so we will explore the solution for Eq. ( 22) in accordance with the following prescribed structure of the method: As the method dictates, substituting ( 23) into ( 22) yields the following nonlinear algebraic and differential equations: Solving the above set of algebraic and differential equations by using Mathematica software package program, we acquire solutions which are representing below: Thus, the variable coefficient Boussinesq system yields the following exact traveling wave solutions: The above mentioned solutions can be transformed into hyperbolic functions, represented as: As A and B are arbitrary functions of time.So it is possible to select their values arbitrarily, which are produc- ing a variety of potential outcomes: Case I: Assuming A = − αa 1 2 β and B = 6 , then solution ( 27) and ( 28) takes the form: Case II: The choice A = 1 2 β and B = 1 αa 1 convert the solutions ( 27) and ( 28) into the form: Vol:.( 1234567890)  28) and ( 28) becomes Case IV: Setting A = αa 1 2 β and B = 6 in solutions ( 27) and ( 28), then wherein ξ = p(t)x + q(t) with p(t) = and q = µ ± 1 6 αa 1 √ β dt.In the same process, for other considerations of the integral constants A and B , more important and logical wave solutions can be obtained but the solutions are not shown in order to keep the size of this article to an elegant volume.

Solutions through the generalized Kudryashov method
Applying the homogeneous balance principle to the leading linear term U ′′ and the non-linear term U 2 in Eq. ( 22) yields: If we choose m = 1 , which gives n = 3.Thus according to the method, the exact solution of (22) takes the following form: Substituting (37) into (22) and using relation Q ′ (ξ ) = Q(ξ )(Q(ξ ) − 1) , we derive the subsequent nonlinear algebraic and differential equations following the method Solving the above set of algebraic and differentials presented in Eqs.(38), using Mathematica software, we found solutions that are presented below: where A stands as an arbitrary constant.
The solutions described above can be expressed in terms of hyperbolic functions as follows: Case I: If we choose A = 1 in the solutions ( 41) and ( 42), then

Case II:
The consideration A = 2 converts the solution ( 41) and ( 42) into the form: Case III: If we set A = −2 , then the solution ( 41) and (42) becomes Case IV: Suppose A = −1 , then the solution ( 41) and ( 42) takes the form: Here ξ = p(t)x + q(t) , where p(t) = and q = µ ± 2 √ βdt.By similar process, it is possible to derive numerous other indispensable and consistent wave solutions through variations in the integral constants A .Nonetheless, in order to maintain elegant brevity of the paper we have chosen not to include these solutions.

Solutions through the modified sine-Gordon expansion method
As the balance number, n = 2 .So based on the method, consider the solution for Eq. ( 22):  22) together with w ′ (ξ ) = sin(w) , then as per the method we can derive the fol- lowing nonlinear algebraic and differential equations: By using the Mathematica software package program to solve the system of algebraic and differential Eqs.(52) presented above, we obtain different sets of solutions that are presented below: Set 1: By using these results, the modified sine-Gordon expansion approach provides the solutions of the timevarying coefficient Boussinesq model as follows: The technique, with the aid of these results, offers the solutions to the time-varying coefficient Boussinesq system as follows: βdt.These results enable the modified sine-Gordon expansion method to present the solutions of the time-varying coefficient Boussinesq equations as follows: βdt.Utilizing these findings, the method yields the solutions to the time-varying coefficient Boussinesq model as follows: In the above sets of solutions, ξ = p(t)x + q(t).

Results and discussion
In this section, we have discussed the exact results obtained by the three methods by considering various parameters through diagrams.We have depicted the profile of soliton solutions obtained by successfully applying the MSE method, the generalized Kudryashov method and the modified sine-Gordon technique to the Boussinesq system with varying coefficients.All the figures provide a remarkable representation of how the soliton solutions of an equation are affected by the variable coefficients of the equation.Selection of different types of functions for the variable coefficients in the system, soliton solutions exhibit a diverse range of waveforms or variations, which are displayed with 3D surfaces and 2D and 3D curves to highlight the variations.It is noticeable from the figures that the variable coefficients of the equation and the wave velocity function continuously change the shape of the wave surface, i.e. the length, height and direction of the wave.To highlight the authenticity of this phenomenon, we have diagrammed the wave velocity solutions ( 29)-( 36), ( 43)-( 50) and ( 53)-( 60) that have physical applications in the real world.

Graphical analysis of solutions: modified simple equation method
We have portrayed the results of the Boussinesq system with time-involving coefficients, marked as ( 29) and ( 30), for two distinct scenarios.In the first case, with = 1 , µ = 0.6 , β dt and in the second case, with = −0.65 , β dt , we have presented these outcomes in Figs. 1(i), (ii) and 2 (i), (ii) respectively, spanning the intervals of −10 ≤ x ≤ 10 and 0 ≤ t ≤ 10 .Additionally, we have included both 2D and 3D solution curves, each associated with specific time value.The solution surfaces for both u and v form the anti-lump shaped waves in the first case, whereas the second case displays irregular periodic solitons with rising wave-amplitudes and wavelengths.On the other hand, the directions of the arrows in the (c) diagrams of all figures reveal how the 2D curves change with time.
Again for = 0.1 β dt , we have plotted the solutions of the time-varying coefficient Boussinesq model, denoted as ( 33) and ( 34), on Figs.5(i), (ii) and 6(i), (ii) respectively.These plots encompassed the spatial domain −10 ≤ x ≤ 10 and the temporal domain 0 ≤ t ≤ 10 , also presented solution curves for distinct time points in both 2D and 3D diagrams.In the first scenario, the solution surfaces represent irregular periodic solitons, while they depict bell-shaped waves in the second scenario.

Graphical analysis of solutions: generalized Kudryashov method
For = 0.25 , µ = 0.5 , α = sec2t , β = tanh 2 t when q = µ − 2 √ βdt and = 1 , µ = −15 , α = 10(t + 1) , when q = µ + 2 √ βdt , the two results ( 43) and ( 44) of the Boussinesq system characterized by time-varying coefficients, have depicted in Figs.7(i), (ii) and 8(i), (ii) over the limitations −10 ≤ x ≤ 10 and 0 ≤ t ≤ 10 , together with 2D and 3D representations of solution curves across different time.In the both cases, the solution surfaces take on distinct forms, with the first case showing double-periodic solitons in both directions and the second case revealing irregular half-range kink with anti-bell and half-range anti-kink with bell shaped solitons along t-axis for u(x, t) and v(x, t) respectively.Within the Boussinesq equations with time-evolving coefficients, we have displayed the two results identified as ( 45) and ( 46) in Figs.9(i), (ii) and 10(i), (ii) throughout the region where −10 ≤ x ≤ 10 and 0 ≤ t ≤ 10 for two specific conditions.In the first condition, with 2 in situation where q = µ − 2 √ βdt and in the second condition, featuring = −0.35, µ = 0.95 , α = sece t/5 , β = 0.5 t/5 whenever q = µ + 2 √ βdt have considered.Additionally these figures show both 2D and 3D solution curves for different values of time.Irregular anti-bell shaped solitons characterize the solution surfaces in the first case, whereas the second case displays irregular double-periodic solitons with respect to both axes.
Finally, we have refrained from showing figures for singular solutions (35), (36), ( 49), ( 50), ( 57)-( 60) of the Boussinesq model, which are mathematically correct but have no physical application in the real world.Although sometimes considering special functions for variable coefficients, some of these solutions behave like non-singular solutions.Thus we have shown only the diagrams of solutions (35) and (36) in Fig. 17(i) and (ii) rather than illustrating all the graphs of those solutions.We have considered = 1 , µ = 0.5 , a 1 = 4 , α = t , β = 3t and taking q = µ − 1 6 αa 1 √ β dt to display the figures for −10 ≤ x ≤ 10 and 0 ≤ t ≤ 10 , where both images reveal singular periodic solitons.Also in these figures the solution curves for different values of time are presented in 2D and 3D diagrams.
From the above discussion it is clear that the applied methods serve as suitable, applicable and very effective tools to find the soliton solutions of the nonlinear evolution models with variable coefficients, which are developed in engineering, physics and applied sciences.

Conclusion
In this study, we have developed substantial techniques for obtaining exact solutions to the Boussinesq systems characterized by the time-varying coefficients.The accessibility of these solutions through the utilization of the modified simple equation, the modified sine-Gordon expansion and the Kudryashov methods open up new paths for understanding and modeling intricate systems with variable coefficients.The explanations and visual presentations of the attained results for different cases increase the accessibility of our findings and enable a more intuitive grasp of the dynamics of the system.This research provides theoretical understanding as well as opportunities for advanced problem-solving technique by bridging the gap between constant and variable coefficient situations in Boussinesq equations.Although these methods have advantages, they encounter limitations when applied to solving equations.These limitations include sensitivity to changes in coefficients, challenges in presenting a physically meaningful interpretation, and the generation of impractical solutions.To address these constraints, there is a need to refine existing methods and explore hybrid approaches.This study might be used https://doi.org/10.1038/s41598-023-50782-1